Page 42 - Advanced Linear Algebra

26 Advanced Linear Algebra

%
&
Example 0.16 The ring 9~ -´%Á &µ of polynomials in two variables and is
not a principal ideal domain. To see this, observe that the set ? of all
polynomials with zero constant term is an ideal in . Now, suppose that is the
?
9
principal ideal ? . Since %Á&  ? ~ º ²%Á&³» , there exist polynomials ²%Á&³
and ²%Á &³ for which
% ~ ²%Á &³ ²%Á &³ and & ~ ²%Á &³ ²%Á &³ ( 0.1)
But ²%Á &³ cannot be a constant, for then we would have ? ~ 9 . Hence,
deg² ²%Á &³³ ‚ and so ²%Á &³ and ²%Á &³ must both be constants, which
(
)
implies that 0.1 cannot hold.
Theorem 0.27 Any principal ideal domain 9 satisfies the ascending chain
condition, that is, cannot have a strictly increasing sequence of ideals
9
? ? ‰ ‰ Ä

where each ideal is properly contained in the next one.
Proof. Suppose to the contrary that there is such an increasing sequence of
ideals. Consider the ideal

<~ ?
which must have the form <~ º » for some  < . Since  ? for some ,

we have ? ? ~ for all ‚ , contradicting the fact that the inclusions are
proper.
Prime and Irreducible Elements
We can define the notion of a prime element in any integral domain. For
Á  9, we say that divides written “ if there exists an %  9 for
)
(
which ~% .
Definition Let be an integral domain.
9
)
1 An invertible element of is called a unit . Thus,  " 9 is a unit if # " ~
9
for some #9 .
)
2 Two elements Á  9 are said to be associates if there exists a unit for
"
which ~" . We denote this by writing — .
)
3 A nonzero nonunit  9 is said to be prime if
“ ¬ “ or “
)
4 A nonzero nonunit  9 is said to be irreducible if
~ ¬ or is a unit
Note that if is prime or irreducible, then so is " for any unit .

"
The property of being associate is clearly an equivalence relation.